While spline functions have been used very successfully for analyzing both exact and noisy data of signals, the technique of Fourier transformation has been the conventional tool for studying the corresponding spectral behavior in the frequency domain. Indeed, in many applications such as image and signal analyses, only spectral information can be observed. However, although the Fourier techniques are very powerful, there is a very serious deficiency of the integral Fourier representation, namely: information concerning the time-evolution of frequencies cannot be obtained with a Fourier transform. The formulation of the Fourier transform requires global information of the function in the time-domain, i.e., from the entire length of the signal in time.
This shortcoming has been observed by D. Gabor, (Gabor, D., "Theory of Communication", JIEE (London) 93 (1946), 429-457) who introduced a time and frequency localization method by applying the Gaussian function to "window" the Fourier integral. Other window functions have been studied since then, and this method is usually called the window Fourier transform or short-time Fourier transform (STFT). Nevertheless, there are still defects in all of the STFT methods, mainly due to very undesirable computational complexity when narrowing the window is required for good localization of signal characteristics and widening the window is required for yielding a more global picture.
The integral wavelet transform (IWT), on the other hand, has the capability of zooming in on short-lived high-frequency phenomena and zooming out for low-frequency observations. Hence, the IWT is suitable for a very wide variety of applications such as radar, sonar, acoustics, edge-detection, etc. This transform has its origin in seismic analysis and the first disclosure of it was in Grossman, A. and J. Morlet, "Decomposition of Hardy Functions Into Square Integrable Wavelets of Constant Shape," SIAM J. Math. Anal. 15 (1984), 723-736. It is based on the simple idea of dilation and translation of the window function which is known as the basic wavelet (B-wavelet).psi.. Dilation corresponds to the change of frequencies, and translation localizes time or position. In addition to being a window function like the STFT, the basic wavelet .psi. satisfies the condition of having zero mean. Hence, since it has somewhat concentrated mass, it behaves like on "small wave", and so the terminology of "wavelet", or "ondelette" in French, is quite appropriate. The zero-mean property of .psi., or equivalently the vanishing of its Fourier transform at the origin .psi.(0)=0, can be weakened a little to .psi.(.omega.)/.vertline..omega..vertline..sup.1/2 being in L.sub.2, a space in which signals or images are finite. The idea and techniques of the IWT can be traced back to Calderon, A. P., "Intermediate Spaces and Interpolation, the Complex Method," Studia Math. 24 (1964), 113-190, which related to singular integral operators. An integral reproducing formula allows us to reconstruct the function from its IWT.
However, such a reconstruction requires global information of the IWT at all frequencies. If the basic wavelet .psi. is orthonormal (o.n.), and this means that {.psi.(.cndot.-n): n.epsilon. } is an o.n. family, then under very mild conditions on .psi., a "wavelet series" can also be used to recover the original function. This observation revolutionizes certain aspects of Harmonic Analysis, in that instead of the Fourier series, which represents periodic functions, the orthogonal series of wavelets represents functions defined on the real line. The importance of this representation is that with a good choice of the basic o.n. wavelet .psi., there is now a representation which localizes both time and frequency.
There are many o.n. wavelets in the literature. The oldest one is the Haar function which cannot be used to localize frequency. Later works which are very influential to the development of the subject of wavelets are the following: Stromberg, J. O., "A Modified Franklin System and Higher-Order Spline Systems of .sup.n As Unconditional Bases For Hardy Spaces," in Conference in Harmonic Analysis in honor of Antoni Zygmund, Vol. II, W. Beckner, et al. (ed.), Wadsworth Math Series, 1983, 475-493; Meyer, Y., "Principe d'Incertitude, Bases Hilbertiennes et Algebres d'Opera Teurs," Seminaire Bourbaki, No. 662, 1985-1986; Lemarie, P. G., "Ondelettes a Localisation Exponentielle," Journal de Math. Pures et Appl. 67 (1988), 227-236; and Battle, G., "A Block Spin Construction of Ondelettes Part I: Lemarie Functions," Comm. Math Phys. 110 (1987), 601-615. The o.n. wavelets .psi. which have the greatest impact to this subject are the compactly supported o.n. wavelets introduced in Daubechies, I., "Orthonormal Bases of Compactly Supported Wavelets," Comm. Pure and Appl. Math. 41 (1988), 909-996.
The inventors have found another approach to reconstruct a function from its IWT, again by a wavelet series. This wavelet .psi. confers a technical advantage in that it does not have to be o.n. as described above, but still gives the same orthogonal wavelet decomposition of the function. The idea is to introduce a dual wavelet .psi. of .psi.. The advantage of this approach is that it gives more freedom to construct wavelets with other desirable properties. Perhaps the most important advantage, at least for many applications to signal analysis, is the property of linear phase, which requires the wavelet to be symmetric or antisymmetric. Since compactly supported o.n. wavelets other than the Haar function cannot be symmetric or antisymmetric (see Daubechies, supra), the inventors give up orthogonality within the same scale-levels in order to construct compactly supported continuous wavelets with linear phase. The extra freedom also allows the inventors to give explicit expressions of the wavelets; and in fact, even compactly supported polynomial spline-wavelets with linear phases having very simple expressions were constructed by the inventors.
The applications of wavelets, whether o.n. or not, go far beyond the IWT and the recovery from IWT. The main reason is that very efficient algorithms, usually called pyramid algorithms, are available. Pyramid algorithms for o.n. wavelets were introduced in Mallat, S., "Multiresolutional Representations and Wavelets," Ph.D. thesis, Univ. of Pennsylvania, Philadelphia, 1988. The inventors derived the pyramid algorithms for nonorthogonal spline-wavelets. All these algorithms yield orthogonal wavelet decompositions and reconstructions in almost real-time. A wavelet decomposition separates and localizes the spectral information in different frequency bands (or octaves), and hence, filtering, detection, data reduction, enhancement, etc., can be easily implemented before applying the wavelet reconstruction algorithm. Both the decomposition and reconstruction algorithms use formulas that describe the intimate relationship between the wavelet of interest and the "spline" function that is used for approximation. The mathematical description of this relationship, called multiresolution analysis, was introduced in Meyer, Y., "Ondelettes et Fonctions Splines," Seminaire Equations aux Derivees Partielles, Centre de Mathematique, Ecole Polytechnique, Paris, France, December 1986) and by Mallat, supra.
There has been a long-felt need in the field for methods and apparatus that are useful for analyzing frequency information of signals and images by which phenomena can be localized, and which are sufficiently computationally simple that they can be performed almost in real time.